If X is a random variable with a Pareto (Type I) distribution,[1] then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by

where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.

When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a log-log plot, the distribution is represented by a straight line.

The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number exceeding , is a Pareto distribution with the same Pareto index but with minimum instead of .

In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.

Pareto distributions

Support

Parameters

Type I

Type II

Lomax

Type III

Type IV

The shape parameter α is the tail index, μ is location, σ is scale, γ is an inequality parameter. Some special cases of Pareto Type (IV) are

The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments are finite for some δ > 0, as shown in the table below, where δ is not necessarily an integer.

Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.[8] This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.[9] However, the 80-20 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his Cours d'économie politique indicates that about 30% of the population had about 70% of the income. The probability density function (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (Note that the Pareto distribution is not realistic for wealth for the lower end. In fact, net worth may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:

The sizes of human settlements (few cities, many hamlets/villages)[10]

The Pareto distribution is a special case of the generalized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the Lomax distribution as a special case. This family also contains both the unshifted and shifted exponential distributions.

The Pareto distribution with scale and shape is equivalent to the generalized Pareto distribution with location , scale and shape . Vice versa one can get the Pareto distribution from the GPD by and .

The "80-20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log4(5) = log(5)/log(4), approximately 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown[15] to be mathematically equivalent:

Income is distributed according to a Pareto distribution with index α > 1.

There is some number 0 ≤ p ≤ 1/2 such that 100p % of all people receive 100(1 − p) % of all income, and similarly for every real (not necessarily integer) n > 0, 100pn % of all people receive 100(1 − p)n percentage of all income.

This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.

This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.

The Lorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve L(F) is written in terms of the PDF f or the CDF F as

where x(F) is the inverse of the CDF. For the Pareto distribution,

and the Lorenz curve is calculated to be

Although the numerator and denominator in the expression for diverge for , their ratio does not, yielding L=0 in these cases, which yields a Gini coefficient of unity. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.

The Gini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (α = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for ) to be

Malik (1970)[17] gives the exact joint distribution of . In particular, and are independent and is Pareto with scale parameter xm and shape parameter nα, whereas has an Inverse-gamma distribution with shape and scale parameters n−1 and nα, respectively.

The characteristic curved 'Long Tail' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a log-log graph, which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for x ≥ xm,

The bounded (or truncated) Pareto distribution has three parameters α, L and H. As in the standard Pareto distribution α determines the shape. L denotes the minimal value, and H denotes the maximal value. (The variance in the table on the right should be interpreted as the second moment).

^ abFeller, W. (1971). An Introduction to Probability Theory and its ApplicationsII (2nd ed.). New York: Wiley. p. 50. "The densities (4.3) are sometimes called after the economist Pareto. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ Ax−α as x → ∞."

^Lomax, K. S. (1954). Business failures. Another example of the analysis of failure data.Journal of the American Statistical Association, 49, 847–852.

^Schroeder, Bianca; Damouras, Sotirios; Gill, Phillipa (2010-02-24). "Understanding latent sector error and how to protect against them". 8th Usenix Conference on File and Storage Technologies (FAST 2010). Retrieved 2010-09-10. We experimented with 5 different distributions (Geometric,Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their ﬁt through the total squared differences between the actual and hypothesized frequencies (χ2 statistic). We found consistently across all models that the geometric distribution is a poor ﬁt, while the Pareto distribution provides the best ﬁt.